Question: In triangle $ABC$, $AB$ is congruent to $AC$, the measure of angle $ABC$ is $72^{\circ}$ and segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. If point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel to segment $BD$, how many isosceles triangles are in the figure shown?

[asy]
size(150);
draw((0,0)--(5,15)--(10,0)--cycle,linewidth(1));
draw((0,0)--(8,6)--(6.5,0)--(9.25,2.25),linewidth(1));
label("B",(0,0),W);
label("A",(5,15),N);
label("D",(8,6),E);
label("E",(7,0),S);
label("F",(9,3),E);
label("C",(10,0),E);

[/asy]
Explanation: Clearly, triangle $ABC$ is isosceles.  This is the first.  We know $\angle ABC = \angle ACB=72^{\circ}$, which tells us that $\angle BAC = 180^\circ-72^\circ-72^\circ=36^\circ$ .  Since segment $BD$ bisects angle $ABC$, the measure of angle $ABD$ is $72^\circ/2=36^\circ$.  Thus, $\angle BAD = \angle ABD$ and $\triangle ABD$ is isosceles.

Since $\triangle ABD$ is isosceles, we see that $m\angle ADB=180^\circ-36^\circ-36^\circ=108^\circ$.  Thus, $\angle BDC=180^\circ-108^\circ=72^\circ$.  Looking at triangle $BDC$, we already know that $\angle DCB=72^\circ=\angle BDC$ degrees, so this triangle is isosceles.

Next, we use the fact that $DE$ is parallel to $AB$.  Segment $BD$ is a transversal, so the alternate interior angles $ABD$ and $BDE$ are congruent.  Thus, $m\angle ABD=m\angle BDE=36^\circ$.  We already knew that $m\angle DBE=36^\circ$ since $BD$ bisects $\angle ABC$.  Thus, the triangle $BDE$ is isosceles.

Looking at angle $EDF$, we can see that $m\angle EDF=180^\circ-m\angle BDA-m\angle BDE=180^\circ-108^\circ-36^\circ=36^\circ$.  We also know that $EF$ is parallel to $BD$, and so the alternate interior angles $\angle BDE$ and $\angle FED$ are congruent.  Thus, $m\angle FED=36^\circ$ and triangle $DEF$ is isosceles.

We have nearly found them all.  We can compute that $\angle EFD=180^\circ-36^\circ-36^\circ=108^\circ$, and so $\angle EFC=180^\circ-108^\circ=72^\circ$ degrees.  From the very beginning, we know that $m\angle ACB =72^\circ$, so $\triangle FEC$ is isosceles.  This makes $m\angle FEC=180^\circ-72^\circ-72^\circ=36^\circ$ degrees, and so $m\angle DEC=36^\circ+36^\circ=72^\circ$.  So, our final isosceles triangle is $DEC$.  We have found a total of $\boxed{7}$ isosceles triangles.